Question

If the system of equations $x-ky+3z=0,$ $2x+ky-2z=0$ and $3x-4y+2z=0$ has non-trivial solutions, then the value of $\frac{10y}{x}$ is equal to

• A. $3$
• B. $-\frac{15}{2}$
• C. $\frac{5}{7}$
• D. $-\frac{5}{7}$

Answer 1

Answer: (B)

For non-trivial solutions,
$\left|\begin{array}{ccc}1& -k& 3\\ 2& k& -2\\ 3& -4& 2\end{array}\right|=0\Rightarrow \left(2k-8\right)+k\left(10\right)+3\left(-8-3k\right)=0$
$\Rightarrow 3k-32=0\Rightarrow k=\frac{32}{3}$
Putting $z=\lambda$ , we get,
$3x-32y=-9\lambda y=\frac{\lambda }{4}$
$3x-4y=-2\lambda x=-\frac{\lambda }{3}$
i.e. $\frac{10y}{x}=-\frac{15}{2}$